For the map 2 decision rule pf alse alarm ln ln 1 f0

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Unformatted text preview: T > t} = P {T1 > t and T2 > t} = P {T1 > t}P {T2 > t} = e 0 −h1 (s)ds e Rt 0 −h2 (s)ds = e− Rt 0 h(s)ds where h = h1 + h2 . Therefore, the failure rate function for the minimum of two independent random variables is the sum of their failure rate functions. This makes intuitive sense; if there is a system that fails when either of one of two components fails, then the rate of system failure is the sum of the rates of component failure. 3.10. BINARY HYPOTHESIS TESTING WITH CONTINUOUS-TYPE OBSERVATIONS (a) h(t) (b) 113 h(t) t t Figure 3.24: Failure rate functions for (a) nearly constant lifetime and (b) an item with a burn-in period (bath tub failure rate function) Two commonly appearing types of failure rate functions are shown in Figure 3.24. The failure rate function in Figure 3.24(a) corresponds to the lifetime T being nearly deterministic, because the failure rate is very small until some fixed time, and then it shoots up quickly at that time; with high probability, T is nearly equal to the time at which the failur...
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